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Homework Statement
Hi everybody! As always, I struggle with my special relativity class and here is a new problem I'd like to have some indications about:
A masspoint m moves in the xyplane under the influence of gravity on a circular path of radius r (see attached pic). Which constraining force ##\vec{z}## must be additionally be acting in the radial direction, so that the masspoint remains on the circular path? You must first establish the equation of motion for the masspoint m and integrate it for the case small angle ##\varphi##.
Homework Equations
Lagrangian mechanics, holonomic constraints
The Attempt at a Solution
So first I wrote the coordinates dependently of each other and in function of ##l##:
##x^2 + y^2 = l^2 \iff x^2 + y^2  l^2 = 0##
If I understand my script correctly, such an equation ##f(x,y,l,t) = 0## means this is a case of holonomic constraint. Right? I am not sure if ##l## should be included in my function. Anyway I can rewrite ##x## and ##y## as
##x = l \cdot \cos \varphi## and ##y = l \ cdot \sin \varphi##
##\implies \dot{x} =  \varphi l \sin \varphi## and ##\dot{y} = \varphi l \cos \varphi##
I can now substitute those values in the Lagrange function to have an equation of motion only depending on ##l## and ##\varphi##:
##L = T  V = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2)  mgh##
## = \frac{1}{2} m\varphi^2 l^2 (\cos^2\varphi + \sin^2\varphi)  mgh##
## = \frac{1}{2} m \varphi^2 l^2  mgh##
That went pretty well until now, but now I'm blocked because of the ##h##. How can I express it in terms of ##l## and ##\varphi##? Geometrically I see the relation but I don't know what ##h## is. Anything I am missing?
Thanks a lot in advance.
Julien.
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